Sunday, September 7, 2014

GEOMETRY – a Jane Schaffer-ish approach to teaching students' first two-column proof

I'm required to teach two-column proofs in Geometry, but having also been trained as an English teacher, this has never seemed like a problem to me. In fact, if anything, the activity I used on Friday seemed to scaffold the process using students' existing knowledge better than anything else I've tried before.

My design process was as follows: because of their ELA and writing backgrounds, students already know far more about constructing an argument in words and statements than we math teachers often give them credit for knowing. All the major writing curricula, such as Jane Schaffer and Six Traits, provide scaffolded methods for teaching students to make claims and to support their assertions with evidence and interpretations that connect that evidence to their claims through interpretive statements. Indeed, the Jane Schaffer method, in particular, has a very lovely scaffolded process (which I've extended in the past) to bridge students' metacognitive processes about their writing, taking them from a place of very concrete thinking to one of considerable abstraction.

 So why not use this same kind of process for proof?

Instead of having students merely "fill in" the reasons for the statements in their first proof (in our curriculum, that's the Midpoint Theorem), I created a task card with instructions and materials for creating a "working poster" (an idea I have adopted from Malcolm Swan) of a two-column proof. They needed to set it up the way we'd done it the day before (two columns, Statements and Reasons), and then they would need to (a) sort their cut-out statements from the task card into a correct order (more than one order is possible), and (b) use their notes and discussions to give the justification or reason that permitted them to make each of these assertions in turn.

The richness of their conversations blew me away. They also confirmed my intuitions that (1) math conversations and projects can indeed draw on students' existing competencies in argumentation that they have developed in their English and Social Studies classes (indeed, many relished the opportunity!), and (b) it is indeed possible to create intellectual need (see Guershon Harel and Dan Meyer) for definitions, postulates, previous theorems, and propositions from algebra through situational motivation.

This activity turned two-column proof into a reasoning and sense-making activity that exposed and built on prior knowledge instead of invalidating it; created what Swan calls "realistic obstacles to be overcome"; turned students' notes into a valued and valuable learning resource; and used higher-order questioning, as opposed to mere recall.
I realize I have not referenced the van Hiele levels here, but that is, in part, because I think I may be kind of bypassing some of their assumptions. I'm not at all sure about this, though, and I would welcome better-informed thoughts and thinking about this in the comments.


Task card for intro to two-column proof: 1-5 intro task Sorti…o 2-Column Proof.pdf

Editable Word doc: 1-5 intro task Sorti…o 2-Column Proof.doc

Original editable Pages doc: 1-5 intro task Sorti…2-Column Proof.pages

Monday, September 1, 2014

What do you do after Formative Assessment reveals a gaping hole in understanding? More Talking Points, of course. :)

My Geometers took the opportunity to inform me through their Chapter 1 exams that they really don't get how angles are named. So this seemed like a perfect opportunity for more Talking Points, of course. :)

This time I'm giving everybody a diagram of a figure that the Talking Points refer to. They will have to do some reasoning about naming angles in order to do the Talking Points. They love doing Talking Points, but they mostly like coming to immediate consensus. Hopefully this will throw a monkey wrench (so to speak) into those works.

Here is the Talking Points file (they print 2-UP) and here is the set of diagrams (they print 6-UP) to use together for this lesson/activity.

More news as it happens!

Sunday, August 31, 2014

PRECALCULUS: Transformations of Functions Speed Dating

I know I am going to take some grief for this statement, but I have learned it to be the truth. There are some things that, once discovered, you simply need to memorize — and practice. This is about mental (and often physical) motor skills. Becoming fluent at some things requires more or less practice for different people, but for most of us, it does require something.

I come by this knowledge honestly.

I have played the piano since I was three or four. In growing as a pianist, there are conceptual learnings, procedural learnings, and contextual learnings. Technique, concepts, and contexts. To play well, you need first to become proficient and fluent technically. That is why there are so many established sets of technical figure exercises: Hanon, Czerny, and many others. The same is true for the violin (Schradieck, Kreutzer, Hrimaly) and every other instrument under the sun. If you want to be capable of playing the advanced works in your instrument's repertoire, your fingers and body need to know how to flow over the keys or strings in every known and familiar situation (scales, arpeggios, finger-crossings) so that you can summon them automatically in your pursuit of unknown and unfamiliar territory in the literature of your instrument.

But working your way through all of the possible finger movements will only make you a master technician. Doing only finger exercises will not give you a very deep understanding of music. It simply builds the finger and muscle memory you will need to face different and difficult technical situations and requirements as you dive deeper and deeper into the piano repertoire.

In other words, technical proficiency is necessary, but not sufficient to become a strong and capable musician. Without technique, you cannot progress.

Beyond technique, there are concepts to master — harmony, counterpoint, and composition, as well as music history. If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire. You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others. Without the foundational concepts, your attempts to interpret and play the repertoire will be incoherent.

Finally, there are contexts — historical, cultural, interpretive. The ultimate context is your instrument's repertoire. Music, like mathematics, is a cultural act. As a musical learner, you need to be mentored into the repertoire. You need to experience other musicians' interpretations and experiences to learn what competent, coherent, and ultimately subtle musical communication sounds, feels, and looks like. And while you are doing this, you also need to be able to explore the repertoire for yourself so that you can find your own way.

It seems to me that this is a similar situation to what we face in mathematics education. Technical proficiency is boring but somewhat mindless. If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind. As music, the technical patterns are boring — up and down, back and forth, crossing and uncrossing, stretching and shifting. But they're necessary to develop a foundation of muscle memory and motor skills, as well as the habits of mind and of practice you will need as you gain proficiency and advance to building the finer and finer skills of musicianship.

So there was a need for teachers to create pieces that are very rich musically but very restricted technically so that they are accessible to new learners. This is why Bach, for example, wrote the pieces in his Anna Magdalena Notebook. They created an on-ramp for his new wife to be included in the family's deep musical conversations. She needed a low barrier-to-entry, high-ceiling on-ramp to sophisticated music. These are also the motivation behind Bach's Two- and Three-Part Inventions. Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education. They are accessible pieces that even the greatest virtuosi find beautiful.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. They are the music of his childhood, but from an advanced standpoint. The recordings are filled with joy and delight. To the dismay of critics, he often hums along with himself as he plays. The critics find this annoying. Personally, I find it inspiring. If Glenn Gould can still delight in playing these pieces, then so can I. It gives me great permission. The recordings and the playing are magical.

In my view, this is how we should be looking at the balance we need to strike in math education. There are technique and concepts and repertoire that learners need. Without strong technique, your understanding of concepts will be shallow. Without context/repertoire, your understanding and practice of mathematics will be joyless and without wonder.

The long-range purpose
of our practice was clearly
stated on the board
So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday. As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus. This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint. The order of operations becomes more sophisticated. "Groupings" replace "parentheses" in your thinking about where to start. Groupings, I warn them, are masters of disguise. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume. They might show up in the form of the absolute value bars or the fraction bar. Don't be fooled, I warn them. Keep your focus on whether you are dealing with transformations of inputs or outputs.

We started simply, using only a single function du jour. On Friday, that was the square root of x. We also restricted our investigation to horizontal and vertical shifts as well as reflections across the x- and y-axis. We were considering the impact on the graphs of basic parent functions as we operated on either the input or the output of the function. We did not multiply by any value other than –1 to start. The Day 1 problem cards are here.  The Day 2 problem cards are also available now (formatted by the amazing Meg Craig - thank you!).

On Friday, each of my 36 students started out with his or her own problem card that contained two related transformations — a shift and a reflection. We organized the speed dating structure, moved our backpacks along the two empty walls, and established our rules of movement (the students along the window side of each row travel, while the hallway-side students stay where they are). If you run into problems, ask the expert in the room on that problem. He or she is sitting directly across from you.
36 precalc students in a state of flow

Trade, analyze, investigate, sketch, discuss. For forty minutes, my Precalculus students lost themselves in analyzing, investigating, sketching, and discussing functional operations on inputs and outputs. Every two minutes, my iPhone timer would go off and I would call out, "Shift!" And all 36 students would trade back their problem cards, while half of them stood up and moved one seat to their right. Then I would reset the timer and they would lose their ego-selves in each new immersion. I loved the hush that fell over the room after everybody settled down into the next round of problems. I eavesdropped on moving and purposeful conversation about inputs and outputs of functions, shifting and reflecting, as they worked collaboratively to help each other and to help themselves attain proficiency. The preciousness of each minute of mathematical conversation was not lost on me.

And I tell you, it was glorious.

Friday, August 22, 2014

WEEK 1 - PROJECT 'MAD MEN' -- Classroom Rules PSA Skits

Leonard Bernstein once said, "To achieve great things, two things are needed — a plan and not quite enough time." I decided to put that principle to the test this first week at my new school by assigning a project on Day 1 that thew together strangers with an absurd but achievable goal: given a particular classroom rule or guideline, create a Public Service Announcement  (i.e., a 30-second "TV commercial" in the form of a skit) whose purpose was to motivate viewers to follow the rules/guidelines for the good of the group.

I created a set-up, instructions, and a rubric for the group project. And my students did not disappoint.

The idea was to get students to think about the consequences of their actions and choices, but their ideas for implementation exceeded even my wildest dreams. Most skits followed a "slice of life" strategy, but the ones that really blew us all away were the ones that parodied existing campaigns.

Two brilliant PSAs started from already-iconic Geico insurance commercials, but the one that left me with tears running down my cheeks was a take-off on Sarah McLachlan's ASPCA spots. The song, "In the arms of the angels..." began playing, and student "Sarah" appears onstage making the exact same kind of appeal she makes in those ads. They had the tone, cadence, and music exactly right, and they clearly understood the emotionally manipulative rhetorical strategy — the seemingly endless list of forms of ignorance designed to eventually provoke self-recognition in almost everyone. Their "mathematical justification" was as follows: the narrator enters and says, "In the past year alone, texting in class tragically cost 5 of Doctor X's students their lives. Remember, think twice before texting in class — there may be fatal consequences for your grade, and for you!

It was pure and inspired genius.

I also loved the spot-on impressions of my teacher persona. One student gave a pitch-perfect parody of my "Function Basics" talk that made me both cringe and laugh my ass off simultaneously.

The best thing about this assignment was that it really pushed the voice of authority downward, into the student community itself. Whatever they made of the experience, they owned it.

I am going to try and remember this for later in the semester, when we've become too routinized.

This is definitely going to be an ongoing part of my repertoire of Day 1 activities. I got through what I neded to,  then gave them the rest of the abbreviated period to collaborate. The time pressure was a thing of art.

It was perfectly imperfect — exactly the way first days ought to be.


Here is the link to a generified Word document that you can customize for your own class:

PROJECT MAD MEN- classroom rules PSA generic.doc

And here are the three sample 30-second PSAs I showed my classes to give them ideas:

'You Lost Your Life!' – game show hosted by the Crash Test Dummies (Since Vince & Larry, the Crash Test Dummies, were introduced to the American public in 1985, safety belt usage has increased from 14% to 79%, saving an estimated 85,000 lives, and $3.2 billion in costs to society)
What could you buy with the money you save?' - throwing things over a cliff (You could purchase TVs, bicycles, and computers with the money most families spend on wasted electricity)
'Five Seconds' – at highway speeds, the average text takes your eyes off the road for 5 seconds (Five seconds is the average time your eyes are off the road while texting. When traveling at 55mph, that's enough time to cover the length of a football field)

Tuesday, August 5, 2014

Edwin Moise on postulates considered as self-evident truths

My favorite book I've read this summer has hands-down been Edwin E. Moise's, Elementary Geometry From An Advanced Standpoint (yes, I know, I'm a nerd). I'm nowhere near finished with it, but in planning for teaching Geometry this year, I've used it a lot to help me think through what we assert and why we assert it.

This section is a beautiful summary of the many wonders and problems with an axiomatic system such as the one we continue to teach. I don't know exactly how I'm going to use this, but it is definitely going to inform my presentation of something.

pp. 282-3


In the time of Euclid, and for over two thousand years thereafter, the postulates of geometry were thought of as self-evident truths about physical space; and geometry was thought of as a kind of purely deductive physics. Starting with the truths that were self-evident, geometers considered that they were deducing other and more obscure truths without the possibility of error. (Here, of course, we are not counting the casual errors of individuals, which in mathematics are nearly always corrected rather promptly.) This conception of the enterprise in which geometers were engaged appeared to rest on firmer and firmer ground as the centuries wore on. As the other sciences developed, it became plain that in their earlier stages they had fallen into fundamental errors. Meanwhile the “self-evident truths” of geometry continued to look like truths, and also continued to seem self-evident.

With the development of hyperbolic geometry, however, this view became untenable. We then had two different, and mutually incompatible, systems of geometry. Each of them was mathematically self-consistent, and each of them was compatible with our observations of the physical world. From this point on, the whole discussion of the relation between geometry and physical space was carried on in quiet different terms. We now thing not of a unique, physically “true” geometry, but of a number of mathematical geometries, each of which may be a good or bad approximation of physical space, and each of which may be useful in various physical investigations. Thus we have lost our faith not only in the idea that simple and fundamental truths can be relied upon to be self-evident, but also in the idea that geometry is an aspect of physics.

This philosophical revolution is reflected, oddly enough, in the differences between the early passages of the Declaration of Independence and the Gettysburg Address. Thomas Jefferson wrote:

  “ . . . We hold these truths to be self-evident, that all men are created equal, that they are endowed by their creator with certain unalienable rights, that among these are Life, Liberty and the pursuit of Happiness . . . . “

The spirit of these remarks is Euclidean. From his postulates, Jefferson went on to deduce a nontrivial theorem, to the effect that the American colonies had the right to establish their independence by force of arms.

Lincoln spoke in a quite different style:

  “Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal.”

Here Lincoln is referring to one of the propositions mentioned by Jefferson, but he is not claiming, as Jefferson did, that this proposition is self-evidently true, or even that it is true at all. He refers to it merely as a proposition to which a certain nation was dedicated. Thus, to Lincoln, this proposition is a description of a certain aspect of the United States (and, of course, an aspect of himself). (I am indebted for this observation to Lipman Bers.)

This is not to say that Lincoln was a reader of Lobachevsky, Bolyai or Gauss, or that he was influenced, even at several removes, by people who were. It seems more likely that a shift in philosophy had been developing independently  of the mathematicians, and that this helped to give mathematicians the courage to undertake non-Euclidean investigations and publish the results.

At any rate, modern mathematicians use postulates in the spirit of Lincoln. The question where the postulates are “true” does not even arise. Sets of postulates are regarded merely as descriptions of mathematical structures. Their value consists in the fact that they are practical aids in the study of the mathematical structures that they describe.

Wednesday, July 30, 2014

"The organism moves towards health" — reflections on TMC14

Everybody is writing blog posts about feeling like a fraud after an amazing experience at Twitter Math Camp 2014. Impostor syndrome. I feel like a fraud too, at least, most of the time, but I am trying to practice refraining from my conditioned habits of reacting automatically and giving in in response to that defense mechanism. I am practicing not-reacting. I am trying to notice the positive energy that is there and to just allow it. I am trying to allow myself to experience myself as a competent, good-enough teacher I have respect for and want to continue to be.
me practicing accepting myself as a competent,
good-enough teacher, seen here
with supportive tweeps & a giant margarita

What worked in the Group Work Working Group session was setting up a structure to sustain that positive energy of presence. Having learned how to do that is a huge gift I have given myself over the past 25 years of dharma practice. It's a "gift" that comes from working very hard at being present and practicing every day, rain or shine, whether I feel like it or not, whether I do it well or do it badly. I follow the three teachings my teacher Natalie Goldberg learned from her teacher Katagiri Roshi: Continue under all circumstances. Don't be tossed away. Make positive effort for the good. I have done that in my practice every single day for over 25 years. It's the one thing I know in my feet that I am good at.
starting with restorative classroom circles

So I decided to bring THAT to Twitter Math Camp this year.

The structure works because it is a structure for teaching and sustaining presence — learning to be present with an open heart. I have dharma sisters and brothers all over the world, but when we practice together online, we practice asynchronously — each of us on our own, in our own lives, in our own homes. When we come together using the asynchronous forum of online communication, we maintain that same structure of presence. Write when you write. Read when you read. Listen when you listen.

No comment.

using the Talking Points structure; no comment
"No comment" is the most important part of the structure, and the hardest part to implement online in a forum like Twitter, which is designed to support comments. "No comment" is about allow there to be space for everyone. It is about all being present together authentically and about staying present with whatever arises. THAT is the thing that most adolescents don't get exposed to in their lives, and it is the thing that can make the greatest possible difference in the quality of their experience — both in the math classroom and everywhere else in their lives.

Most people in our culture don't have a lot of experience in being present and staying present. It takes an enormous amount of energy to learn how to stay present and not flinch. But do that with anything you love and you will have a magical experience. Do that with math, and you'll unlock the treasures of your amazing human mind. But being present with others in a big open space is hard. At first it can be scary. It's very naked. That is why the structure of "no comment" is so important. It helps to create a shared space of emotional safety. It gets everybody focused on their own stuff and supports dropping the "act" you bring to most in-person interactions. That's why it's good to do so right from the start. It's about reframing our conditioned habits of personality.

Very quickly, the timed structure and the practice of "no comment" makes the practice of presence very freeing. You begin to relax into that big open space. You become curious. Your defenses soften. You begin to notice the interesting patterns of your own mind. Best self and worst self. Curious self and bored self. Zen mind and monkey mind. Defense mechanisms, such as snark.

collaborative mathematics
using the Talking Points structure; no comment
The practice of "no comment" creates a space in which the authentic thoughts of your own amazing human mind can arise and step forward. And we honor that process by persisting in not-commenting as we continue.

Natalie describes this process as stepping forward with your own mind.

Once you get a taste for being present, you'll naturally begin to crave it more. That is something I count on in my classroom management practice. Fred always said, "The organism moves towards health." That is one of his greatest teachings for me. "The organism moves towards health" means that, in the process of growing up, we all fall away from the naturally sane and healthy patterns of our organism. "Fight or flight" is a falling away from the natural discharge cycle of "rest and digest" we experienced as infants. When you're hungry, you eat. When you're tired, you sleep. Fred said there is a deeper wisdom inside us that is always available for us to tap back into. It's like an underground stream that is part of our psychological and emotional water table. When we practice being present through structures like Talking Points or meditation or writing practice, it feels like a homecoming — a homecoming to a natural state that is healthy and inquisitive and curious to see what will happen next. It is a natural reconnection with our own inner growth mindset that is our birthright — not some artificial fantasy state we impose on students from without by telling them to have one.

assigning competence after group work & 
observation; still no comment
A growth mindset is just the psyche's way of attuning to the fundamental idea that our organism moves naturally in the direction of health if we will let it — if we can get out of its way and allow it to unfold as it needs to. Allowing means learning to refrain from interfering with that natural movement, and so we use structures that make it manageable for ordinary human beings like us to access the extraordinary ocean of intellectual and creative possibility that is mathematics.

Ten minutes at a time is about what I can muster, I have learned over the years.
Kate test-driving a geometry task using the
Talking Points structure; still no comment
In my experience teaching meditation and writing practice and other structures that cultivate presence, I have found it is about what most people can handle. Ten minutes of Talking Points, no comment — GO. Ten minutes of writing practice — keep your hand moving, no comment, GO. Ten minutes of mathematical conversation, no comment, GO. Learning how to be present with the big, scary openness of not-knowing is no small thing. That is why we zone out, check our phones a hundred times an hour, play video games, watch TV, assault-eat, numb out, zone out, distract ourselves. We all crave the real stuff, but connecting with it feels like sticking a butter knife into the electrical socket. So we break it into more manageable chunks. We set a limit for ourselves and dive in for a limited period. We practice being present for ten minutes at a time. And then we give ourselves and our students a break. It helps us build our tolerance for the intensity of presence and it builds our courage to come back and try it again the next time.

Natalie says that monkey mind is the guardian at the gate, protecting the treasures of our heart and strengthening us for the challenge of opening ourselves to presence and to Big Mind. The structure of "no comment" makes it feel safe for us to touch in to that fire at the center of our being. It helps us to close the gap between what we THINK we've been doing and what we have ACTUALLY done. For me, it's about strengthening students' courage to open their hearts to contact with their amazing mathematical minds — with what my friend Max Ray of The Math Forum at Drexel calls their "mathematical imaginations." We math teachers know the secret that everyone has this mathematical imagination. Our greatest challenge is to get students to trust that they have it too and can access it safely and reliably.


Read Taming Your Gremlin by Rick Carson.

Tuesday, July 22, 2014

TMC #14 GWWG – Annotated References: the research on explicit teaching of exploratory talk

OK, last summarizing post before I have to get packing.

Here is the background research on ‘exploratory talk. Once again, please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

Shell Centre,  MAP PD: Students Working Collaboratively (PD Module 5)
An adequate introduction, with pointers to some of the major reseach on exploratory talk in the math classroom. PD module on using talk in the math classroom. There are two parts to download: the overview and the “handouts for teachers.” The Handouts for Teachers is more useful than the overview, but you kind of need both to get started.

Neil Mercer and Steve Hodgkinson, editors, Exploring Talk in School
The richest single source of research on cultivating exploratory talk, though many of the chapters I found most useful are available on the internet.

Douglas Barnes, “Exploratory Talk for Learning” (chapter 1 in Exploring Talk in School, but downloadable PDF here)
Barnes is the research who originally pioneered the concept of exploratory talk as distinguished from cumulative talk and disputational talk. Everybody else’s work builds on his.

Neil Mercer and Lyn Dawes, “The value of exploratory talk” (chapter 4 in Exploring Talk in School, but downloadable PDF here)
An explanation of the need for differing kinds of talk in the classroom (asymmetrical teacher-student talk as well as symmetric student-student or peer talk). Makes a strong case for doing the work to intentionally raise the level of symmetrical talk to make group work effective and meaningful.

Lyn Dawes, The Essential Speaking and Listening,  Chapter 2: “Talking Points” (downloadable PDF here). 
Overview of her original strategy for using the “Talking Points” activity structure to encourage students to develop what Barnes referred to as “an open and hypothetical style of learning.” This is what I used as the basis for developing my own version of the Talking Points activity.

Thinking Together
How can children be explicitly enabled to use talk more effectively as a tool for reasoning? The Thinking Together program was founded to address this specific question. A freely available, evidence-based interventional program of three talk lessons plus supporting materials, developed by University of Cambridge Faculty of Education researchers (led by Mercer and Dawes) as “a dialogue-based approach to the development of children’s thinking and learning.” It has been widely implemented in the U.K. as a means of cultivating exploratory talk in the classroom. Their materials are targeted at elementary children, but can be adapted for older students as well.

Sylvia Rojas-Drummond and Neil Mercer, “Scaffolding the development of effective collaboration and learning”
Summary of a collaboration between Mexican and British researchers that documents the effectiveness of using exploratory talk strategies to improve collaborative reasoning and learning in the group work-centered classroom.

Neil Mercer and Claire Sams, “Teaching children how to use language to solve maths problems”
Summary of the research behind the interventional program called Thinking Together  and how to proactively make talk-based group activities more effective in developing students’ mathematical reasoning, understanding, and problem-solving.